Exploring dissipative sources of non-Markovian biochemical reaction systems

Many biological processes including important intracellular processes are governed by biochemical reaction networks. Usually, these reaction systems operate far from thermodynamic equilibrium, implying free-energy dissipation. On the other hand, single reaction events happen often in a memory manner, leading to non-Markovian kinetics. A question then arises: how do we calculate free-energy dissipation (defined as the entropy production rate) in this physically real case? We derive an analytical formula for calculating the energy consumption of a general reaction system with molecular memory characterized by nonexponential waiting-time distributions. It shows that this dissipation is composed of two parts: one from broken detailed balance of an equivalent Markovian system with the same topology and substrates, and the other from the direction-time dependence of waiting-time distributions. But, if the system is in a detailed balance and the waiting-time distribution is direction-time independent, there is no energy dissipation even in the non-Markovian case. These general results provide insights into the physical mechanisms underlying nonequilibrium processes. A continuous-time random-walk model and a generalized model of stochastic gene expression are chosen to clearly show dissipative sources and the relationship between energy dissipation and molecular memory.

Figure 1

Shown is an example for energy dissipation decomposition. (a) Schematic description of the decomposition, showing that the total entropy production rate is split into two parts: one for the equivalent Markovian system, the other due to molecular memory characterized by nonexponential waiting-time distributions. (b) Three possible decomposition patterns for nontrivial entropy production rate, where BDB is an abbreviation for broken detailed balance; DTD for direction-time dependence; DTI for direction-time independence; and DB for detailed balance.

Figure 2

Energy dissipation in a CTRW model. (a) The global scenario for dependence of the equivalent dissipation ${\dot{S}}_E$ on both ratios ${\lambda }_{12}/\phantom{{\lambda }_{12}{\lambda }_{13}}{\lambda }_{13}$ and ${\lambda }_{23}/\phantom{{\lambda }_{23}{\lambda }_{21}}{\lambda }_{21}$. (b) The global scenario for dependence of the memory dissipation ${\dot{S}}_M$ on both ratios ${\lambda }_{12}/\phantom{{\lambda }_{12}{\lambda }_{13}}{\lambda }_{13}$ and ${\lambda }_{23}/\phantom{{\lambda }_{23}{\lambda }_{21}}{\lambda }_{21}$. (c) the global scenario for dependence of the memory dissipation ${\dot{S}}_M$ on both ratio ${\lambda }_{12}/\phantom{{\lambda }_{12}{\lambda }_{13}}{\lambda }_{13}$ and transition probability $p_2$ if the system is in DB. (d) Special case of (c), which shows dependence of the memory dissipation ${\dot{S}}_M$ on ratio ${\lambda }_{12}/\phantom{{\lambda }_{12}{\lambda }_{13}}{\lambda }_{13}$ for three fixed transition probabilities $p_2$. The transition probabilities are set as: $p_1=0.6, p_2=0.7, p_3=0.8$ for (a) and (b), and $p_1=1-p_2, p_3=0.5$ for (c) and (d), and other parameters are set as: ${\lambda }_{13}=10, {\lambda }_{21}=20, {\lambda }_{31}={\lambda }_{32}=50$. In addition, ${\lambda }_{23}=20$ is set for (c) and (d).

Figure 3

(a) Schematic description of a generalized gene expression model, where the reaction processes are characterized by waiting-time distributions as indicated. (b) Schematic description of state transitions with six possible patterns. The top row figures show that the current state is at OFF state whereas the bottom row figures show that the current state is at ON state. In all cases, $\mathit{n}, {\mathit{n}}^{\prime }$, and ${\mathit{n}}^{\prime \prime }$ denote the current state, next state, and last sate, respectively. Note that the current state $\mathit{n}$ may be the last state ${\mathit{n}}^{\prime \prime }$.

Figure 4

Influence of molecular memory on mRNA expression and memory difference. (a) Stationary mRNA distributions in Case 1; (b) Influence of memory index $L_1$ on mean mRNA level in Case 1; (c) Influence of $L_1$ on mean memory difference in Case 1; (d) Stationary mRNA distributions in Case 2; (e) Influence of memory index $L_2$ on mean mRNA level in Case 2; (f) Influence of $L_2$ on mean memory difference in Case 2. (g) Stationary mRNA distributions in Case 3; (h) Influence of memory index $L_3$ on mean mRNA level in Case 3; (i) Influence of $L_3$ on mean memory difference in Case 3. Parameters are set as ${\mu }_0=1,\mu =20,{\lambda }_1=3L_1, {\lambda }_0=5L_2, \delta =L_3.$

Figure 5

Effects of molecular memory on energy dissipation. Heatmaps (a), (d), respectively, show the equivalent dissipation and the memory dissipation as a function of both memory index $L_1$ and inactivation rate ${\lambda }_0$. Heatmaps (b), (e), respectively, show the equivalent dissipation and the memory dissipation as a function of both memory index $L_2$ and activation rate ${\lambda }_1$. Heatmaps (c), (f), respectively, show the equivalent dissipation and the memory dissipation as a function of both memory index $L_3$ and inactivation rate ${\lambda }_0$. The setting of parameter values is the same as in Fig. 4.

Figure 6

Effects of molecular memory on mean dissipation. Heatmaps (a), (b), respectively, show the mean equivalent dissipation and the mean memory dissipation as a function of both memory index $L_1$ and inactivation rate ${\lambda }_0$. Heatmaps (d), (e), respectively, show the mean equivalent dissipation and the mean memory dissipation as a function of both memory index $L_2$ and activation rate ${\lambda }_1$. Heatmaps (g), (h), respectively, show the mean equivalent dissipation and the mean memory dissipation as a function of both memory index $L_3$ and inactivation rate ${\lambda }_0$. (c), (f), (i) Dependence of the mean equivalent dissipation and mean memory dissipation on $L_1$ for a fixed ${\lambda }_0=5, L_2$ for a fixed ${\lambda }_1=3,$ and $L_3$ for a fixed ${\lambda }_0=5,$ respectively. The setting of other parameter values is the same as in Fig. 4.